Glossary of differential geometry and topology

Short description: Mathematics glossary

This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:

* Glossary of general topology
* Glossary of algebraic topology
* Glossary of Riemannian and metric geometry.

See also:

* List of differential geometry topics

Words in italics denote a self-reference to this glossary.

## A

* Atlas

## B

* Bundle – see fiber bundle.

* basic element – A basic element x with respect to an element y is an element of a cochain complex (C*,d) (e.g., complex of differential forms on a manifold) that is closed: dx=0 and the contraction of x by y is zero.

## C

* Chart

* Cobordism

* Codimension – The codimension of a submanifold is the dimension of the ambient space minus the dimension of the submanifold.

* Connected sum

* Connection

* Cotangent bundle – the vector bundle of cotangent spaces on a manifold.

* Cotangent space

## D

* Diffeomorphism – Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeomorphism – if both f:M->N and its inverse f−1:N->M are smooth functions.

* Doubling – Given a manifold M with boundary, doubling is taking two copies of M and identifying their boundaries. As the result we get a manifold without boundary.

## E

* Embedding

## F

* Fiber – In a fiber bundle, π:E->B the preimage π−1(x) of a point x in the base B is called the fiber over x, often denoted Ex.

* Fiber bundle

* Frame – A frame at a point of a differentiable manifold M is a basis of the tangent space at the point.

* Frame bundle – the principal bundle of frames on a smooth manifold.

* Flow

## G

* Genus

## H

* Hypersurface – A hypersurface is a submanifold of codimension one.

## I

* Immersion

* Integration along fibers

## L

* Lens space – A lens space is a quotient of the 3-sphere (or (2n \+ 1)-sphere) by a free isometric action of Z – k.

## M

* Manifold – A topological manifold is a locally Euclidean Hausdorff space. (In Wikipedia, a manifold need not be paracompact or second-countable.) A Ck manifold is a differentiable manifold whose chart overlap functions are k times continuously differentiable. A C∞ or smooth manifold is a differentiable manifold whose chart overlap functions are infinitely continuously differentiable.

## N

* Neat submanifold – A submanifold whose boundary equals its intersection with the boundary of the manifold into which it is embedded.

## O

* Orientation of a vector bundle

## P

* Parallelizable – A smooth manifold is parallelizable if it admits a smooth global frame. This is equivalent to the tangent bundle being trivial.

* Poincaré lemma

* Principal bundle – A principal bundle is a fiber bundle P->B together with an action on P by a Lie group G that preserves the fibers of P and acts simply transitively on those fibers.

* Pullback

## S

* Section

* Submanifold – the image of a smooth embedding of a manifold.

* Submersion

* Surface – a two-dimensional manifold or submanifold.

* Systole – least length of a noncontractible loop.

## T

* Tangent bundle – the vector bundle of tangent spaces on a differentiable manifold.

* Tangent field – a section of the tangent bundle. Also called a vector field.

* Tangent space

* Thom space

* Torus

* Transversality – Two submanifolds M and N intersect transversally if at each point of intersection p their tangent spaces Tp(M) and Tp(N) generate the whole tangent space at p of the total manifold.

* Trivialization

## V

* Vector bundle – a fiber bundle whose fibers are vector spaces and whose transition functions are linear maps.

* Vector field – a section of a vector bundle. More specifically, a vector field can mean a section of the tangent bundle.

## W

* Whitney sum – A Whitney sum is an analog of the direct product for vector bundles. Given two vector bundles α and β over the same base B their cartesian product is a vector bundle over B×B. The diagonal map B->B×B induces a vector bundle over B called the Whitney sum of these vector bundles and denoted by α⊕β.

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